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The finite-frequency dynamics of the $^4$He superfluid phase transition canbe formulated in terms of the response of thermally excited vortex loops to anoscillating flow field. The key parameter is the Hausdorff fractal dimension$d_H$ of the loops, which affects the dynamics because the frictional force ona loop is proportional to the total perimeter $P$ of the loop, which varies as$P \sim a^{d_H}$ where $a$ is the loop diameter. Solving the 3D Fokker-Planckequation for the loop response at frequency $\omega $ yields a superfluiddensity which varies at $T_{\lambda}$ as $\omega^{1/(d_H -1)}$. This power-lawvariation with $\omega$ agrees with the scaling form found by Fisher, Fisher,and Huse, since the dynamic exponent $z$ is identified as $z = d_H-1$. Floryscaling for the self-avoiding loops gives a fractal dimension in terms of thespace dimension $d$ as $d_H = (d+2)/2$, yielding $z = d/2 = 3/2$ for d = 3, incomplete agreement with dynamic scaling.

Finite-frequency dynamics of vortex loops at the $^4$He superfluid phase transition